2023/03/21 by Christoforos Neofytidis, Neofytidis, Christoforos, Hongbin Sun +7 · 1 citation
Mathematics · Computer Science · #Advanced Topology and Set Theory #Rings, Modules, and Algebras #Digital Image Processing Techniques
paper · pdf · doi:10.48550/arxiv.2303.11922
The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b∈ D(M), then ab∈ D(M); a set A⊆\mathbb Z so that ab∈ A for each a,b∈ A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set [NWW] and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds [CMV]. We show the following: (i) Not every multiplicative set A containing 0,1 is a self-mapping degree set. (ii) For each n∈\mathbb N and k≥3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and [CMV], every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n≠ 1,2,4,5.